**2.2 Modeling of the battery-pack's complex impedance**

To carry out EIS tests, an impedance analyzer is generally used. This device generates a frequency sweep signal and measures the voltage and current in each test point. As a result, the complex impedance is calculated. Because most of commercial impedance analyzer generates AC signals less than 100 mA (suitable for cell testing), in this work, this signal is amplified and controlled by means of the experimental test bench deeply explained in [26].

EIS tests have been performed at different SOC values (20, 40, 60, 80, and 90% SOC) to analyze the effects of these SOC variations. The frequency sweep has been set from 1 mHz to 5 kHz (typically test range), with an AC ripple of 5 A (10% *Imax*). **Figure 5** shows the Nyquist plots of the EIS results, which has been used to analyze the impedance behavior of the tested battery module. In this graph, the real part of the complex impedance (Z') is represented along the x-axis and the imaginary part (Z") along the y-axis. The capacitive behavior corresponds to negative values of Z" and the inductive behavior to the positive ones. In this way, it is easy to identify the parameters of the electrical circuit. According to EIS results, the impedance

**Figure 4.** *OCV-SOC characteristic.*

#### **Figure 5.** *EIS tests result.*

of the pack tested shows a capacitive behavior from 1 mHz to 316 Hz. From this value, the pack impedance corresponds to an ideal inductance. The differences in these plots reflect that SOC variations affect the capacitive behavior from 1 mHz to 4 Hz. For low frequencies, the Nyquist plots show that both real (Z') and imaginary

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**Figure 6.**

results and presented in **Table 3**.

*Time constants associated with pack impedance.*

**2.3 Hybrid model of the battery-pack**

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency…*

(Z") parts of the impedance decrease with frequency, drawing a line with a slope of almost 45°. From 18 mHz to 4 Hz, the plots present semicircular shapes, whose diameter diminishes with increasing SOC. For medium frequencies (4 to 316 Hz), the impedance corresponds to a semicircle of constant diameter. To simulate these capacitive behaviors, several *RC* networks connected in series can be used [20, 21].

To analyze the influence of the impedance components on the dynamic response of the battery-pack, **Figure 6** shows the EIS result at 40% SOC. The results analysis allows to associate the most relevant time constants with the impedance behavior of the battery-pack. Most of the dynamic applications of the batteries (load/frequency control or renewable generation support) have their time constants from 1 mHz to 316 Hz; for this reason, the inductive behavior can be neglected. In this frequency range, the *RC* networks that reproduce the impedance behavior can be used to determine the different time constants that affect the dynamic response of the tested battery-pack. These time constants *(τ1, τ2, τ3*) are calculated from EIS test

As a result of these combined time- and frequency-domain tests, an electrical circuit has been determined. Also, the model includes an integration current SOC estimator, to guarantee that the parameters of the electrical circuit simulate the dynamic behavior of the battery-pack for different SOC conditions, as shown in **Figure 7**. The inputs of the model are the initial value of SOC (*SOC0*), the batterypack capacity (*Cn*), and the current (*ipack*). The output corresponds to the voltage response of the battery-pack (*upack*), which is calculated by Eq. (2), where *uRo*, *uc1*,

The value of equivalent resistance of the battery-pack is *Ro* ≅ 39 mΩ.

*DOI: http://dx.doi.org/10.5772/intechopen.88718*

*Hybrid Modeling Procedure of Li-Ion Battery Modules for Reproducing Wide Frequency… DOI: http://dx.doi.org/10.5772/intechopen.88718*

**Figure 6.** *Time constants associated with pack impedance.*

*Research Trends and Challenges in Smart Grids*

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**Figure 5.** *EIS tests result.*

**Figure 4.**

*OCV-SOC characteristic.*

of the pack tested shows a capacitive behavior from 1 mHz to 316 Hz. From this value, the pack impedance corresponds to an ideal inductance. The differences in these plots reflect that SOC variations affect the capacitive behavior from 1 mHz to 4 Hz. For low frequencies, the Nyquist plots show that both real (Z') and imaginary (Z") parts of the impedance decrease with frequency, drawing a line with a slope of almost 45°. From 18 mHz to 4 Hz, the plots present semicircular shapes, whose diameter diminishes with increasing SOC. For medium frequencies (4 to 316 Hz), the impedance corresponds to a semicircle of constant diameter. To simulate these capacitive behaviors, several *RC* networks connected in series can be used [20, 21]. The value of equivalent resistance of the battery-pack is *Ro* ≅ 39 mΩ.

To analyze the influence of the impedance components on the dynamic response of the battery-pack, **Figure 6** shows the EIS result at 40% SOC. The results analysis allows to associate the most relevant time constants with the impedance behavior of the battery-pack. Most of the dynamic applications of the batteries (load/frequency control or renewable generation support) have their time constants from 1 mHz to 316 Hz; for this reason, the inductive behavior can be neglected. In this frequency range, the *RC* networks that reproduce the impedance behavior can be used to determine the different time constants that affect the dynamic response of the tested battery-pack. These time constants *(τ1, τ2, τ3*) are calculated from EIS test results and presented in **Table 3**.
